How can I model a spherical tesselation such as this one in blender?
Create a cube, in edit mode hit W and subdivide it 4 times. Select the center loops of the main faces:



Then, invert the selection, with Ctrl+I and delete those faces:



After that, select only the faces that are connecting the loops, turn on Proportional Editing, with O, change the pivot point to Individual Origins and set the Transformation Orientation to Normal; hit R, to rotate, then Z twice to limit rotation along the normal, and input 45 as the rotation angle and adjust falloff with the middle mouse wheel; hit Enter when finished.





Select all the faces, hit Shift+Alt+S and enter 1 to make the shape spherical.

Final result with creased edges, then solidify modifier and subdivision surface: